1. Bayesian Averaging of Classifiers and the Overfitting Problem Rayid Ghani ML Lunch – 11/13/00

2. BMA is a form of Ensemble Classification Set of Classifiers Decisions combined in ”some” way Unweighted Voting Bagging, ECOC etc. Weighted Voting Weight  accuracy (training or holdout set), LSR (weights  1/variance) Boosting

3. Bayesian Model Averaging All possible models in the model space used (weighted by their probability of being the “Correct” model) Posterior of a model = Prior * Likelihood given data Optimal given the correct model space and priors Claimed to obviate the overfitting problem by cancelling the effects of different overfitted models (Buntine 1990)

4. BMA - Training prior likelihood noise model posterior ignored If h predicts correct class c i for x i otherwise OR

5. BMA - Testing Pure Classification Model P(c|x,h)=1 for the class predicted by h for x OR Class Probability Model

6. Problems How to get the priors How to get the correct model space Model space too large – approximation required Model with highest posterior, Sampling (Imp sampling,MCMC)

7. BMA of Bagged C4.5 Rules Bagging is an approximation of BMA by importance sampling where all samples are weighed equally Weighting the models by their posteriors should lead to a better approximation Experimental Results Every version of BMA performed worse than bagging on 19 out of 26 UCI datasets Posteriors skewed – dominated by a single rule model – model selection rather than averaging

8. Experimental Results Every version of BMA performed worse than bagging on 19 out of 26 UCI datasets Best performing BMA was uniform class noise and pure classification Posteriors skewed – dominated by a single rule model even though error rates were similar Model selection rather than averaging?

9. Bagging as Imp Sampling Want to approximate Sample according to q(x) and compute the average of f(x)p(x)/q(x) for points x sampled Each sampled value will have weight p(x)/q(x)

10. BMA of various learners RISE Rule sets with partitioning 8 databases from UCI BMA worse than RISE in every domain Trading Rules If the s-day moving average rises above the t-day one, buy; else sell Intuition (there is no single right rule so BMA should help) BMA similar to choosing the single best rule

11. Likelihood of a model increases exponentially with with s/n Small random variation in the sample can sharply increase the likelihood of a model

12. Overfitting in BMA Issue of overfitting is usually ignored (Freund et al. 2000) Is overfitting the explanation for the poor performance of BMA? Preferring a hypothesis that does not truly have the lowest error of any hypothesis considered, but by chance has the lowest error on training data. Overfitting is the result of the likelihood’s exponential sensitivity to random fluctuations in the sample and increases with # of models considered

13. To BMA or not to BMA? Net effect will depend on which effect prevails? Increased overfitting (small if few models are considered) Reduction in error obtained by giving some weight to alternative models (skewed weights => small effect) Ali & Pazzani (1996) report good results but bagging wasn’t tried Domingos (2000) used bootstrapping before BMA so the models were built from less data

14. Spectrum of ensembles Asymmetry of weights Overfitting BaggingBagging BoostingBoosting BMABMA

15. Bibliography Domingos Freund, Mansour, Schapire Ali, Pazzani